The Eye of the Azure is the title given to the individual who holds the true power of the Azure and in turn, the Azure Grimoire.
It is unknown how one can actually become the "Eye" as the only known Eyes are Noel Vermillion/Mu-12 and the original Boundary Interface Prime Field Device, but it is implied that one cannot be human as the power of the Azure would corrupt their body and mind (similar to Arakune). Being the Eye grants the bearer numerous abilities, one of which is, of course, being able to use the power of the Azure but it also grants one to see distant and previous memories by touching the individual. This is seen with Carl Clover, Tsubaki Yayoi, Hazama, and Lambda-11. Due to comments dropped by Rachel, it is also possible that becoming the Eye would make that being an immortal similar to her.